Using Subjective Expectations to Forecast Longevity: Do Survey Respondents Know Something We Don’T Know?

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C.

Using Subjective Expectations to Forecast Longevity: Do Survey Respondents Know Something We Don’t Know?

Maria G. Perozek 2005-68

NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Using Subjective Expectations to Forecast Longevity: Do Survey Respondents Know Something We Don’t Know? Maria G. Perozek1 14 December 2005

Abstract: Future old-age mortality is notoriously difficult to predict because it requires not only an understanding of the process of senescence, which is influenced by genetic, environmental and behavioral factors, but also a prediction of how these factors will evolve going forward. In this paper, I argue that individuals are uniquely qualified to predict their own mortality based on their own genetic background, as well as environmental and behavioral risk factors that are often known only to the individual. Using expectations data from the 1992 HRS, I construct subjective cohort life tables that are shown to predict the unusual direction of revisions to U.S. life expectancy by gender between 1992 and 2004; that is, the SSA revised up male life expectancy in 2004 and at the same time revised down female life expectancy, narrowing the gender gap in longevity by 25 percent over this period. Further, the subjective expectations of women suggest that female life expectancies produced by the Social Security Actuary might still be on the high side, while the subjective life expectancies for men appear to be roughly in line with the 2004 life tables.

1The author thanks Michael Palumbo, Lise Vesterlund, and Jim Walker for their helpful comments on a previous draft of this paper. The opinions expressed here are those of the author and not necessarily those of the Board of Governors of the Federal Reserve System or its staﬀ. Please address correspondence to Maria Perozek, Federal Reserve Board, mail stop 97, 20th and C Streets NW, Washington DC 20551, USA. Tel: 202 452-2692. Fax: 202 728-5889. Email: [email protected]. 1 Introduction

The 20th century witnessed unprecedented improvements in life expectancy: In the United States, life expectancy at birth rose from 47 years in 1900 to 77 years in 2000 (National Center for Health Statistics, 2004).1 Although most demographers agree that mortality rates will continue to decline in the 21st century, there is little consensus on how fast and for how long they will continue to fall (e.g. Vaupel and Lundstrom, 1994; Lee, 2003). The answers to these questions are at the heart of some of the most important issues in the economics of aging, including income adequacy in retirement, and the solvency of the social security system. Many mortality forecasts are based on extrapolations of historical data. However, extrap- olating historical trends may be misleading. For example, simple extrapolative procedures fail to incorporate information about the causes of mortality change over time. This paper provides a somewhat unorthodox alternative to using historical data to project the future path of mortality risk. The method proposed here uses data on individual subjective expectations of survival to construct subjective life tables for a particular cohort. This method has an important advantage over extrapolative methods in that subjective expectations incorporate current and future expected values of variables that influence mortality risk, such as exercise, diet and smoking habits, as well as current and expected advances in medical technology. As much of this information is private, individuals are uniquely qualified to as- sess how these factors will influence their personal mortality risk, which is a function of their medical history, current health status, and family history. By aggregating these individual forecasts of mortality risk across persons in a given cohort, we obtain a subjective cohort life

1Cutler and Meara (2004) provide an excellent overview of the causes underlying mortality improvements

over the twentieth century.

1 table that incorporates causal mechanisms implicitly and does not explicitly depend on ad hoc historical trends. The purpose of this paper is to explore the mortality forecasts implied by the subjective expectations of a cohort of individuals in the Health and Retirement Study (HRS) in 1992. There are three main findings: First, the subjective life tables differ significantly from the life tables put together by the Social Security Actuary (SSA) in 1992 and subsequently revised in 2004, and the deviations from the life table differ significantly by gender. In particular, the subjective life expectancies estimated for men are higher than SSA life tables predict, while the subjective life expectancies for women are a good bit lower. Second, these subjective life tables suggest a further narrowing of the gender gap in longevity in coming decades, with men living longer and women dying earlier than is currently predicted by the SSA. Part of this narrowing has already been reflected in revisions to the SSA life tables between 1992 and 2004 in which male life expectancies were revised up and female life expectancies were revised down. In essence, the subjective expectations data from 1992 predicted the direction of revisions to the SSA life tables between 1992 and 2004. The subjective expectations data also suggest that there should be a further narrowing in the gender gap in longevity for these cohorts that is not yet reflected in the SSA life tables. Finally, I demonstrate that the validity of the subjective survivor functions depends crucially on the functional form that governs changes in mortality after age 85. I show that different functional forms result in significantly different life expectancies, largely stemming from the shape of the survivor function beyond age 85; nevertheless I argue that the main findings of the paper are robust to these assumptions. The paper proceeds as follows. Section 2 describes the unique expectations data available in the Health and Retirement Study. The third section demonstrates how these data can be used to construct individual-specific survivor functions, which are then aggregated using

2 population weights for a cohort of men and women in the HRS. The fourth section discusses the resulting subjective life tables and compares their mortality predictions to the life tables produced by the Social Security Actuary in 1992 and then again in 2004. The ﬁnal section oﬀers concluding remarks and directions for future research.

2 The Health and Retirement Study

The data used in this analysis are from the ﬁrst wave of the Health and Retirement Study (HRS). Initial interviews were conducted in 1992 and provide detailed information on the health status and socio-economic status of a nationally representative sample of persons aged 51 to 61 and their spouses.2 A total of 12,652 individuals were included in the ﬁnal HRS sample in 1992. Variables of particular importance for this paper include subjective expectations of survival to age 75 and age 85, as well as indicators of the age and sex of the respondent. This paper uses the HRS data on survival expectations to generate sequences of survival probabilities for each individual in the sample. In particular, respondents were asked to answer the following questions:

I would like to ask you about the chance that various events will happen in the future. Using any number from zero to ten, where zero equals absolutely no chance and 10 equals absolutely certain, what do you think are the chances that you will live to be 75 or more? And how about the chances that you will live to be 85 or more?

2Detailed documentation of the HRS is available in Juster and Suzman (1995).

3 When the responses to this question are divided by 10, they can be thought of as probabilities of surviving to age 75 and 85, hereafter referred to as P75 and P85. Hurd and McGarry (1995) suggest that, on average, the subjective survival probabilities are internally consistent: The probability of living to age 75 is greater than or equal to the probability of living to age 85. They also demonstrate that the subjective probabilities are externally consistent in that they covary in reasonable ways with other variables such as health status, and that they are on average in the ballpark of the 1988 life table probabilities. Hurd and McGarry (2002) also demonstrate that subjective survival probabilities have predictive validity; that is, those who survived between waves 1 and 2 of the HRS reported signiﬁcantly higher probabilities of survival in 1992 than those who died.3 For this analysis, I focus on men and women aged 52 and 57 for comparison to the 1940 and 1935 birth-year cohort life tables, respectively. I drop 2.5 percent of the observations that had reported subjective probabilities that were not internally consistent, i.e., the subjective probability of living to 85 was strictly greater than the subjective probability of living to age 75. Persons who report the same value for P75 and P85 are retained in the sample, but their reported probabilities are altered somewhat in order to estimate the parameters of the survivor functions.4

3More generally, there is an interesting literature on the validity and interpretation of subjective expectations data, including Bassett and Lumsdaine (2001), Bernheim (1989, 1990), Dominitz (1998), Dominitz and Manski (1997), Hamermesh (1985), and Manski (1990).

4For practical reasons documented in Appendix A, the subjective expectations data are adjusted for respondents who report P75 = P85, and for respondents who report survival probabilities of zero or one.

4 3 Using Expectations Data to Predict Mortality

3.1 Constructing Individual Subjective Survivor Functions

This section describes how the subjective expectations data from the HRS can be used to generate a sequence of subjective survival probabilities–or a subjective survivor function–for each individual in the sample. The basic method proposed here involves fitting a survivor function through the points P75 and P85 on the subjective survivor function. Note that this method is very different in spirit from an alternative method proposed by Gan, Hurd and McFadden (2003) that uses a Bayesian update model to construct individual subjective survivor functions.5 For the purposes of this paper, I maintain the assumption that the individual survivor functions can be approximated by a particular functional form. Two functional forms are commonly used in survival analysis–the Weibull distribution and the Gompertz distribution. The Weibull distribution has been used extensively to model the lifetimes of manufactured goods, as well as the lifetimes of insects, animals, and people (Lawless, 1982). The popularity of the Weibull distribution in survival analysis owes, in part, to its flexibility in allowing decreasing or increasing hazard functions. Another attractive feature of the Weibull is that the mean and variance of the distribution have closed-form solutions (Lawless, 1982). The Gompertz distribution has been popular among demographers because its double exponential form has been thought to reflect the underlying process of aging that leads to

5Gan, Hurd and McFadden (2003) use data from the Asset and Health Dynamics of the Oldest-Old

(AHEAD), which is representative of the population aged 70 and older, to estimate individual speciﬁc survivor functions. As a result, a direct comparison of the mortality forecasts from the diﬀerent methods for a constant cohort are unavailable.

5 death. Despite studies which show that the Gompertz model may not accurately character- ize mortality risk among the oldest-old–i.e. mortality hazards do not appear to continue to increase at the same exponential rate among the oldest-old–this distribution is still widely used and accepted (Economos, 1982; Wilson, 1994). The Weibull and Gompertz distributions each have two parameters, which implies that they are exactly identified given two points on the survivor function, P75 and P85. However, when P75 is sufficiently close to P85, the exactly-identified survivor functions are implausibly flat, yielding unreasonably high probabilities of survival in old age for a significant fraction of the sample. To induce the estimated survival probabilities to be close to zero in extreme old age, I introduce a third point on the subjective survivor function to which most respondents would not likely object. In particular, I set the probability of living to age 110 near zero according to the simple conditional probability:

Pi(110|agei)= Px(110|85, agei) ∗ Pi(85|agei) | SSA lifetable{z } | HRS{z } where the ﬁrst term on the right-hand side is the probability of surviving to age 110 given that a person survives to age 85 for x ∈{male, female}. This term is calculated separately for men and women from the SSA cohort life tables (Bell, Wade and Goss, 1992). The second term represents the subjective probability of living to age 85 given that the respondent is agei in 1992 (P85). The general strategy of this methodology is to estimate the parameters of the survivor function given P75, P85 and P110 using nonlinear least squares (NLLS). In particular, I assume that:

6 Pi,t = Si,t(αi, βi)+ ǫi,t

where Pi,t is the probability that individual i lives to age t, and Si,t is a general representation of a two-parameter survivor function. The error term ǫi,t is assumed to be i.i.d, mean zero,

and homoskedastic. The NLLS estimators are the values of αi and βi that minimize the following expression:

2 Pt∈{75,85,110}[Pi,t − Si,t(αi, βi)]

Two sets of parameter estimates are calculated, the ﬁrst under the assumption that the

W W survivor function takes the form of the Weibull survivor function (αi , βi ), which is given by:

W W W W t−agei βi S (α , β )= exp[−( W ) ] i,t i i αi

and the second under the assumption that it takes the form of the Gompertz survivor function, which is deﬁned as:

G G G G αi G S (α , β )= exp[ G exp(β (t − agei))] i,t i i βi i

This estimation procedure assumes that each individual faces a unique sequence of survival probabilities that are generated from an individual-speciﬁc Weibull (Gompertz) survivor function. Further, each individual reports their survival probabilities with error. Under

7 these assumptions, NLLS will provide unbiased and eﬃcient estimates of the underlying parameters of the survivor function for each individual. As we show in the next section, an aggregate life table can be computed by applying population weights to the individual survivor functions.6

3.2 Constructing Subjective Cohort Life Tables

The two sets of NLLS estimates are used to generate a series of subjective survival probabilities– a Weibull and a Gompertz–for each person in the sample. To generate a representative cohort life table, these subjective probabilities are multiplied by the HRS person-level weight and summed for each age-gender group. That is, the N sample members who are age X in 1992 N (call it the age-X1992 cohort) represent a total population cohort of Pi=0 Wi persons, or the sum of the person-level weights (Wi), in 1992. Going forward, the number of persons from N the age-X1992 cohort expected to be alive at age X + t is given by Pi=0 WiSi,t. This calcu- lation gives the number of persons in the age − X1992 cohort that are expected to be alive at every age x>X1992, or in nomenclature of the life tables, lx. Once we obtain lx for each age x, we can deduce all of the other life table functions as follows:

6Alternatively, if one assumed that each individual in a given age-sex cohort actually faced the same

Weibull survivor function, and reported those probabilities with error, one could estimate the parameters of the aggregate survivor function by weighted nonlinear least squares on the entire cohort. Estimates of aggregate Weibull parameters using this method yielded life expectancies that were a bit higher for the

1940 cohort, but the main results of this paper still hold. Given the variation in risk factors and responses regarding expectations of survival, we maintain the assumption that each individual faces a person-speciﬁc survivor function, and we construct the life tables accordingly.

8 dx = lx − lx+1 q dx x = lx

lx+lx+1 Lx = 2 ω Tx = Pt=0 Lx+t e Tx x = lx

Conceptually, the life table age-speciﬁc death rate, qx, is simply a count of the number of persons who die between age x and age x + 1, dx, divided by the number of persons alive at age x, lx. Note that this function explicitly accounts for the selection of healthier individuals into older age groups, as persons with higher mortality risk are more likely to die at younger

ages and are therefore less likely to be included in the denominator lx as x increases. As is customary, these estimates assume that deaths are distributed uniformly over the

year, so that the average number of persons alive between time t and t+1 is equal to Lx,

which is the midpoint of lx and lx+1. The sum of Lx+t from t = 0 to ω, where ω is the maximum possible age, gives the total number of person-years lived by the cohort over its

lifetime (Tx). Life expectancy is derived by dividing the total number of person-years lived 7 by the cohort (Tx) by the total number of people alive at t=0 (lx).

4 Results

4.1 Men, 1940 Cohort

Although life tables could be constructed for all age cohorts, this paper presents selected life table functions only for the cohorts that align with the 1940 and 1935 cohort life tables

7These basic life table functions are described in more detail in Pollard, Yusuf and Pollard (1991).

9 published in 1992; that is, men and women aged 52 and 57 in 1992, respectively.8 Table 1 presents the survival probabilities derived from the Weibull and Gompertz distributions for the 1940 male cohort; for comparison, the table also shows the life table estimates that were published by Social Security in 1992 and 2004. The table shows that the Gompertz survivor function is quite a bit flatter than the Weibull; as illustrated in figure 1, the Gompertz survival probabilities are significantly lower than the Weibull probabilities through age 75, then a bit higher through age 95, before dropping much faster after age 95. It appears that the survival probabilities from the Gompertz survivor function are too low at younger ages, perhaps indicating that the Gompertz distribution is not appropriate. Indeed, Wilson (1994) notes that for human survivor functions, there appears to be a shift in the exponential parameter at older ages; that is, mortality does not increase at the same exponential factor over the entire length of life, it likely decelerates in old age.9 Not surprisingly, the life expectancy is higher in the subjective cohort life table derived from the Weibull relative to the Gompertz life table: The Weibull estimate of life expectancy at age 52 is 28.2 years while the Gompertz life expectancy is 26.5 years. Table 1 shows that these Gompertz and Weibull life expectancy estimates are between 1/2 year and 2-1/4 years higher, respectively, than the life expectancy of 25.9 years published in the 1940 cohort life table from 1992–the year that these subjective expectations data were gathered. If these subjective life tables had been taken seriously in 1992, they may have suggested that the life expectancy estimates for this cohort were too low. Indeed, as shown in the top row of the right-hand side columns of table 1, the Social Security Actuary revised up their estimates

8Complete subjective cohort life tables are included in Appendix B.

9Vaupel et al. ( 2004) investigate the possibility that mortality rates actually decline beyond a certain

age–a phenomenon they term negative senescence.

10 of life expectancy for this cohort by a significant margin of .8 years when it reestimated the 1940 cohort life table in 2004. The revision to the life table estimates are shown in more detail in figure 2, which compares the SSA life tables published in 1992 and 2004 to the Weibull subjective survivor function. The results show that the revisions to the 1940 cohort life table in 2004 pushed the survival probabilities from the cohort life table into closer alignment with the subjective survivor function at almost every age up through the early 90s. Moreover, the figure shows that the Weibull estimates track the 2004 life table estimates almost exactly up through age 80, at which point the subjective survivor function diverges from the SSA life table. In particular, the Weibull survivor function has a much fatter right tail, implying the probability of surviving to older ages is a good bit higher than the current life table estimates predict. As discussed below, the key to estimating life expectancy for this age group lies in the assumptions underlying mortality forecasts at ages 85 and up.

4.2 Women, 1940 Cohort

Table 2 presents the survival probabilities derived from the Weibull and Gompertz distributions for the 1940 female cohort. As shown in figure 3, the comparison of the survivor functions from these two distributions mirror the male 1940 cohort: The Gompertz survivor function is flatter and has lower probabilities of survival after age 95 than the Weibull. In addition, the Weibull life expectancy of 29.9 years–shown in the first row of table 2–is about 2 years longer than the Gompertz life expectancy, implying a range of subjective life expectancies between 27.9 and 29.9 for this cohort. That said, the subjective life expectancies for women and men in the 1940 cohort compare very differently with the SSA life tables. While the subjective life expectancies for men were

11 higher than the SSA life tables, the subjective life expectancies for women in this cohort are a good bit lower. The four columns to the right in table 2 show that the life expectancy for women in this cohort was 30.9 years according to the SSA life tables published in 1992–or about 1 to 3 years higher than the subjective life expectancies. Therefore, the subjective expectations from 1992 suggest that the SSA life expectancies from 1992 were too high. Remarkably, the SSA revised down its estimate of female life expectancy for this cohort to 30.4 years as of 2004–a downward revision of 1/2 year. As shown in figure 4, the Weibull survivor function looks quite different from the life table survivor functions, with lower probabilities of survival at younger ages and higher probabilities of survival for the oldest- old. It is interesting to note that the reductions in life expectancy between the 1992 and 2004 life table estimates owes largely to a reduction in survival probabilities among those 85 and older. In contrast, the lower life expectancy implied by the Weibull stems from lower survival probabilities through about age 90. Although the functional forms given by the Weibull and the Gompertz are very important for determining the sequence of survival probabilities, the general results hold even when looking at the raw weighted responses to the expectations questions. Tables 3 and 4 show the weighted means of the actual survey responses of P75 and P85 for men and women age 52 and 57 in 1992. These figures differ a bit from the predicted values based on the fitted Weibull and Gompertz survivor functions presented in tables 1 and 2, but yield the same basic conclusions. That is, men in both age groups had much higher estimates of their probability of surviving to age 85 than indicated in the life tables published in 1992. And, upward revisions to the SSA life table probabilities in 2004 halved the difference between SSA estimates and the subjective estimates. Meanwhile, women reported subjective probabilities of survival that were lower than the life tables by a good margin for both P75 and P85. In this case, the life table probability of

12 living to 75 was revised up slightly between 1992 and 2004, while the probability of living to 85 and beyond was revised down. Taken together, the life expectancy for women in both cohorts was revised down 1/2 year in each cohort, moving the life table estimate closer to the subjective life expectancy estimates.

4.3 The Gender Gap

These results for men and women indicate that the gender differential in mortality risk was perceived in 1992 to be declining faster than predicted by the Social Security Actuary at that time. As shown in table 5, the life tables from 1992 predicted that the difference between female and male life expectancy was about 5 years for both cohorts. By 2004, revisions to the male and female life tables for these cohorts reduced the gender gap to about 3-3/4 years–a 25 percent downward revision in just over a decade. The lower panel of table 5 shows that the implied longevity differential from the subjective life tables, which range between 1-1/2 and 2 years, is still quite a bit lower than the 2004 life table estimates. These expectations suggest that the mortality differential between men and women in these cohorts could narrow even further. The bottom line is that the subjective cohort life tables, which are based only on data collected in 1992, predicted revisions to the SSA cohort life tables between 1992 and 2004. These included upward revisions to male life expectancy, and downward revisions to female life expectancy, implying a narrowing of the gender gap.

4.4 Functional Form Assumptions

A key maintained assumption in this analysis is that the Weibull surivivor function ﬁtted through three points on the subjective survivor function can yield a meaningful sequence

13 of survival probabilities for each person in the HRS sample. The Weibull, as noted above, appears to yield higher probabilities of living beyond age 95, even when the estimated life expectancy is lower than the life table estimates, e.g. for women shown in Table 4. This section explores the implications of using the Weibull to fit survivor functions, particularly for comparison to the life tables produced by the Social Security Actuary. One might argue that the Weibull is not be flexible enough to capture the shape of the human survivor function; in particular, the fat right tail associated with the Weibull is inappropriate and may be driving the results described above. To explore the importance of functional form assumptions, I fit Weibull functional forms to the SSA cohort life tables from 1992 and 2004 using the same 3 points of the survivor function used in the subjective life tables: P75, P85, and P110. Figure 6 shows that for men, the Weibull functional form predicts higher survival probabilities both before age 75 and after age 90 than the life table probabilities.10 As a result, life expectancies derived from these Weibull estimates are higher; indeed, as shown at the bottom of table 3, male life expectancies are roughly 1 year to 1-1/2 years higher than those computed by SSA. However, this transformation in effect makes the life table probabilities more directly comparable to the subjective life tables, and the results are somewhat reassuring. The subjective life expectancy was still quite a bit higher (1-1/4 years) than the fitted Weibull life table life expectancy from 1992, but matched the fitted Weibull life table estimate for 2004. Thus, the main result still holds: Subjective life expectancies from 1992 predicted an upward revision to male life expectancies between 1992 and 2004.11 For reference, the SSA life table probabilities were also fitted to the Gompertz functional

10This is also true for the older 1935 male cohort.

11The same basic result obtains for men aged 57 in 1992.

14 form, shown in the last row of table 3. The Gompertz, which does a much better job of fitting the right tail of the survivor function, implies a lower life expectancy than the Weibull that is closer to the actual life table life expectancies. As with the Weibull, the basic results described above continue to hold. The results for women shown in table 4 are similar, only in this case, the life expectancy estimates from both the Weibull and Gompertz are higher than the subjective life expectancies calculated for each cohort. Nevertheless, the main result holds: SSA fitted life expectancies were revised down about the same amount as the actual life expectancies, and the subjective life expectancy is still lower than both the actual and fitted SSA life table estimates from 2004. The memo items in table 5 show that the diminution in the gender gap in longevity is highly stable across the different fitted and actual life expectancy values. Although the fitted life table estimates of the gap are slightly higher than the actual, the percent reduction from 1992 to 2004 is 30 percent–roughly the same as the actual reduction between the 1992 and the 2004 life tables.

5 Concluding Remarks

The Weibull and the Gompertz diﬀer dramatically in their implications about mortality risk at very old ages, with the Weibull implying higher rates of survival for the oldest old than the Gompertz. Because the 1940 and 1935 cohorts have only just now (in 2005) reached the ages 65 and 70, respectively, their mortality experience at the oldest ages has not yet been realized. Moreover, there is a wide range of opinion about the pace of future mortality improvements at very advanced ages. In one camp are those who believe that the pace of future improvements will slow because we are nearing a biological limit to human life

15 expectancy (Olshansky and Carnes, 2001). In the other camp are those who believe that we have not yet come close to the biological limit of human life expectancy (see, e.g. Oeppen and Vaupel, 2002).12 What is not disputed is that past forecasts of mortality improvements have been far too conservative (Oeppen and Vaupel, 2002) and that assumptions about future old age mortality are vital to estimating the expected longevity of current and future cohorts. Future old-age mortality is very difficult to predict because it requires not only an understanding of the process of senescence, which is influenced by genetic, environmental and behavioral factors, but also a prediction of future medical advances as well as other important environmental variables. In this paper, I suggest that individuals have a unique understanding of their own individual aging processes conditional on their own genetic background and environmental and behavioral risk factors. Given this private information, individuals form expectations about future survival probabilities that may provide additional information to demographers and policymakers in their challenge to predict future mortality. I find that expectations elicited in 1992 predicted the unusual direction of revisions to U.S. life expectancy by gender between 1992 and 2004; that is, male life expectancy was revised up and female life expectancy was revised down. The subjective expectations of women suggest that female life expectancies produced by the Social Security Actuary might still be on the high side, while the subjective life expectancies for men appear to be roughly in line with the 2004 life tables.

12For example, estimates from a risk factor simulation model developed by Manton, Stallard and Tolley

(1991) suggest that life expectancy at birth could be dramatically higher than the U.S. life tables currently predict.

16 Table 1 Subjective Life Tables vs. Cohort Life Tables (Social Security Actuary) Men Aged 52 in 1992

Subjective Life Tables 1940 Cohort Life Table Age Weibull Gompertz Published in 1992 Published in 2004 Survival Life Survival Life Survival Life Survival Life Probability Expectancy Probability Expectancy Probability Expectancy Probability Expectancy

S x e x S x e x S x e x S x e x

52 1 28.2 1 26.5 1 25.9 1 26.7

55 0.976 25.9 0.946 24.9 0.976 23.5 0.977 24.3

65 0.848 18.9 0.773 19.4 0.853 16.1 0.86 16.8

75 0.647 13.2 0.589 13.9 0.618 10.1 0.647 10.6

85 0.366 9.3 0.378 8.8 0.287 6.0 0.332 5.7

95 0.146 6.5 0.153 4.4 0.053 3.5 0.053 3.0

105 0.029 5.8 0.008 1.5 0.002 2.2 0.001 1.9

115 0.006 4.8 0 1.0 0 1.3 0 1.1 Table 2 Subjective Life Tables vs. Cohort Life Tables (Social Security Actuary) Women Aged 52 in 1992

S x e x S x e x S x e x S x e x

52 1 29.9 1 27.9 1 30.9 1 30.4

55 0.979 27.5 0.946 26.4 0.986 28.3 0.986 27.8

65 0.876 20.1 0.783 20.9 0.909 20.3 0.911 19.7

75 0.696 13.9 0.621 15.0 0.747 13.5 0.752 12.7

85 0.411 9.9 0.427 9.5 0.487 7.8 0.47 7.0

95 0.178 6.8 0.193 4.8 0.153 4.2 0.118 3.5

105 0.038 5.8 0.015 1.7 0.011 2.5 0.005 2.0

115 0.008 4.8 0 1.2 0 1.3 0 1.1 Table 3: Weighted Means of P75 and P85, Men (standard errors in parentheses) Age 52 in 1992 Age 57 in 1992 subjective subjective expectation life table 1992: life table 2004: expectation life table 1992: life table 2004: (n=395) 1940 cohort 1940 cohort (n=391) 1935 cohort 1935 cohort

P75 0.635 0.618 0.646 0.618 0.633 0.657 (.015) (.015)

P85 0.377 0.287 0.332 0.366 0.288 0.327 (.015) (.015) life expectancy 25.9 26.7 21.6 22.2 life expectancy from fitted Weibull 28.2 26.9 28.2 23.2 22.5 23.5 life expectancy from fitted Gompertz 26.5 25.6 26.9 21.4 22.4

Table 4: Weighted Means of P75 and P85, Women (standard errors in parentheses) Age 52 in 1992 Age 57 in 1992 subjective subjective expectation life table 1992: life table 2004: expectation life table 1992: life table 2004: (n=472) 1940 cohort 1940 cohort (n=415) 1935 cohort 1935 cohort

P75 0.678 0.747 0.752 0.653 0.761 0.764 (.013) (.015)

P85 0.428 0.487 0.47 0.43 0.489 0.468 (.014) (.015) life expectancy --- 30.9 30.4 --- 26.4 25.8 life expectancy from fitted Weibull 29.9 32.5 32.1 24.8 27.9 27.4 life expectancy from fitted Gompertz 27.9 31.2 30.8 26.6 26.1 Table 5: Differences in Life Expectancy by Gender: Female Life Expectancy less Male Life Expectancy

Age 52 Age 57

Cohort Life Table, 1992 5 4.8 Cohort Life Table, 2004 3.7 3.6 percent change -26% -25%

Subjective Expectations (Weibull) 1.7 1.6

Memo: Cohort Life Table 1992, fitted Weibull 5.6 5.4 Cohort Life Table 2004, fitted Weibull 3.9 3.9 percent change -30% -28%

Cohort Life Table 1992, fitted Gompertz 5.6 5.2 Cohort Life Table 2004, fitted Gompertz 3.9 3.7 percent change -30% -29% Figure 1 Subjective Survivor Functions: Men Aged 52 in 1992 (1992 HRS)

1.0000

0.9000

0.8000

0.7000

0.6000

Weibull 0.5000 Gompertz

0.4000 Probability of Survival 0.3000

0.2000

0.1000

0.0000

2 5 8 1 4 7 0 3 6 9 2 5 8 1 4 7 0 3 6 9 2 5 8 5 5 5 6 6 6 7 7 7 7 8 8 8 9 9 9 0 0 0 0 1 1 1 1 1 1 1 1 1 1 Age Figure 2 Survivor Functions for Men Aged 52 in 1992

1.0000

0.9000

0.8000

0.7000

0.6000 Weibull (HRS 1992) 0.5000 SSA cohort life table (1992) SSA cohort life table (2004)

0.4000 Probability of Survival

0.3000

0.2000

0.1000

0.0000

2 5 8 1 4 7 0 3 6 9 2 5 8 1 4 7 0 3 6 9 2 5 8 5 5 5 6 6 6 7 7 7 7 8 8 8 9 9 9 0 0 0 0 1 1 1 1 1 1 1 1 1 1 Age Figure 3 Subjective Survival Functions: Women Aged 52 in 1992 (1992 HRS)

1.000

0.900

0.800

0.700

0.600

Weibull 0.500 Gompertz

0.400 Probability of survival 0.300

0.200

0.100

0.000

2 5 8 1 4 7 0 3 6 9 2 5 8 1 4 7 0 3 6 9 2 5 8 5 5 5 6 6 6 7 7 7 7 8 8 8 9 9 9 0 0 0 0 1 1 1 1 1 1 1 1 1 1 Age Figure 4 Survivor Functions for Women Aged 52 in 1992

1.000

0.900

0.800

0.700

0.600 Weibull (HRS 1992) 0.500 SSA cohort life table (1992) SSA cohort life table (2004) 0.400 Probability of survival 0.300

0.200

0.100

0.000

2 5 8 1 4 7 0 3 6 9 2 5 8 1 4 7 0 3 6 9 2 5 8 5 5 5 6 6 6 7 7 7 7 8 8 8 9 9 9 0 0 0 0 1 1 1 1 1 1 1 1 1 1 Age Figure 5 Weibull Subjective Survivor Functions by Gender, Age 52 (1992 HRS)

1.0000

0.9000

0.8000

0.7000

0.6000

Men 0.5000 Women

0.4000 Probability of survival 0.3000

0.2000

0.1000

0.0000

2 5 8 1 4 7 0 3 6 9 2 5 8 1 4 7 0 3 6 9 2 5 8 5 5 5 6 6 6 7 7 7 7 8 8 8 9 9 9 0 0 0 0 1 1 1 1 1 1 1 1 1 1 Age Figure 6 Fitted Weibull vs. Published Life Table SSA 1940 Male Cohort (2004)

1 0.9 0.8 0.7 0.6 Fitted Weibull 0.5 Published life table 0.4 0.3 Probability of Survival 0.2 0.1 0

2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 5 5 6 6 6 7 7 8 8 8 9 9 0 0 0 1 1 1 1 1 1 1 Age A Appendix

Because of the form of the survivor function, the Weibull parameters are undeﬁned for

persons who report P75 = P85. However, as Hurd and McGarry (1995) note, a respondent

who reports P75 close to P85 may be conveying valuable information regarding the perceived ﬂatness of the subjective survivor function, and it would be unfortunate to be forced to exclude such a large and potentially interesting segment of the sample. The format of the expectation questions in the ﬁrst wave of the HRS requires respondents to round survival probabilities to the nearest tenth. As a result, it is reasonable to assume that the “true”

expectation lies in some interval around P75 = P85, i.e. P75 ∈ [P75-.05, P75+.05] and P85

∈ [P85-.05, P85+.05]. Hence, to retain nearly one-third of the sample who report P75 = P85,

I reassign the probability of living to 75 equal to the upper bound of the interval (P75+.05)

and set the probability of living to 85 to the lower bound of the interval(P85 - .05). For example, a person who reported P75 = P85 = .5 would be reassigned P75 = .55 and P85 = .45. This assignment rule imposes the maximum distance allowed within the interval, thereby implying more credible Weibull estimates. In addition, in order to estimate the Weibull, probabilities of zero and 1 are reassigned

.01 and .99, respectively. If P75 = P85 = .99, then P85 is set to .95, and if P75 = P85 = .01,

then P75 is set to .05. To check the robustness of the results to these assumptions, I started with the unadjusted reported survival probabilities and followed the same procedure for constructing subjective survivor functions.13 Of the 472 women aged 52 in 1992, 137–or about 30 percent–reported

P75 = P85, and of that group, 1/3 reported both probabilities equal to 1, 1/5 reported both

13 All adjustments were removed except for the cases where P75 = P85 = 0, which cannot be estimated via the Weibull without some adjustment.

17 probabilities equal to 0, and 1/5 reported both probabilities equal to .5. The life expectancies derived from the unadjusted survival probabilities are generally higher than the adjusted life expectancies, particularly where the probabilities of living to 75 and 85 are close to or equal to 1. For example, if reported probabilities of living to 75 and 85 are both equal to 1, the unadjusted Weibull life expectancy is roughly 10-1/2 years higher than the adjusted life expectancy for 52 year olds (54.7 years vs. 44.2 years). In the aggregate, the unadjusted subjective life expectancies for the 1940 cohort were about 1 year higher for both men and women than the adjusted life expectancies, bringing the women more in line with the SSA life tables, but exacerbating the diﬀerence for men, and leaving the gender gap about unchanged. Therefore, these results would still predict a narrowing of the gender gap, although they would suggest that men will live even longer relative to the SSA life tables than reported in this paper. Although the path of the survival probabilities generated by the unadjusted variables is fairly similar to that derived from the adjusted probabilities through about age 95, the unadjusted probabilities of survival are much higher between age 95 and 110 before dropping down because of the higher life expectancies and lower variances estimated for those optimistic respondents who reported that they were certain to live to age 85. The unusually high probabilities of survival at these ages lead me to favor the adjusted life table estimates reported in this paper. The problems with estimating these survivor functions point to the importance of understanding mortality rates among the oldest old, for which we have no subjective data beyond age 85. For future work in this area, it would be useful to have another point on the subjective survivor function to work with, perhaps the probability of surviving to age 95.

18 Appendix B Table B.1 Table B.2 Men Aged 52 in 1992 Women Aged 52 in 1992 x qx lx dx Lx Tx ex x qx lx dx Lx Tx ex

52 0.0064 1015813 6532 1012547 28648439 28.2 52 0.0054 1147588 6247 1144464 34330656 29.9 53 0.0086 1009281 8634 1004963 27635893 27.4 53 0.0072 1141341 8217 1137232 33186191 29.1 54 0.0097 1000646 9661 995816 26630929 26.6 54 0.0081 1133123 9134 1128557 32048959 28.3 55 0.0105 990985 10393 985788 25635114 25.9 55 0.0087 1123990 9752 1119114 30920403 27.5 56 0.0112 980591 10993 975095 24649325 25.1 56 0.0092 1114237 10235 1109120 29801289 26.7 57 0.0119 969598 11534 963832 23674231 24.4 57 0.0097 1104003 10654 1098676 28692169 26.0 58 0.0126 958065 12055 952037 22710399 23.7 58 0.0101 1093348 11054 1087821 27593493 25.2 59 0.0133 946010 12581 939719 21758362 23.0 59 0.0106 1082294 11464 1076562 26505672 24.5 60 0.0141 933429 13129 926864 20818643 22.3 60 0.0111 1070831 11905 1064878 25429110 23.7 61 0.0149 920299 13710 913444 19891779 21.6 61 0.0117 1058926 12395 1052728 24364232 23.0 62 0.0158 906590 14330 899425 18978334 20.9 62 0.0124 1046530 12952 1040054 23311504 22.3 63 0.0168 892260 14994 884763 18078909 20.3 63 0.0131 1033579 13588 1026785 22271449 21.5 64 0.0179 877266 15706 869413 17194147 19.6 64 0.0140 1019991 14317 1012832 21244665 20.8 65 0.0191 861560 16466 853327 16324734 18.9 65 0.0151 1005674 15150 998099 20231832 20.1 66 0.0204 845094 17272 836458 15471407 18.3 66 0.0163 990524 16097 982475 19233733 19.4 67 0.0219 827822 18122 818761 14634948 17.7 67 0.0176 974427 17163 965845 18251258 18.7 68 0.0235 809700 19010 800195 13816187 17.1 68 0.0192 957263 18349 948089 17285413 18.1 69 0.0252 790690 19928 780726 13015992 16.5 69 0.0209 938914 19648 929090 16337325 17.4 70 0.0271 770762 20866 760329 12235266 15.9 70 0.0229 919266 21047 908742 15408235 16.8 71 0.0291 749895 21810 738990 11474937 15.3 71 0.0251 898219 22520 886959 14499492 16.1 72 0.0312 728085 22745 716713 10735947 14.7 72 0.0274 875699 24035 863682 13612533 15.5 73 0.0335 705340 23655 693513 10019234 14.2 73 0.0300 851664 25549 838890 12748851 15.0 74 0.0360 681686 24524 669423 9325721 13.7 74 0.0327 826116 27017 812607 11909962 14.4 75 0.0386 657161 25345 644489 8656298 13.2 75 0.0355 799099 28399 784899 11097355 13.9 76 0.0413 631816 26116 618758 8011809 12.7 76 0.0385 770700 29669 755865 10312455 13.4 77 0.0443 605701 26850 592275 7393051 12.2 77 0.0416 741031 30831 725615 9556590 12.9 78 0.0476 578850 27577 565062 6800775 11.7 78 0.0450 710200 31924 694238 8830975 12.4 79 0.0514 551274 28333 537107 6235713 11.3 79 0.0487 678276 33006 661773 8136737 12.0 80 0.0557 522940 29143 508369 5698606 10.9 80 0.0528 645270 34091 628225 7474964 11.6 81 0.0607 493797 29973 478810 5190238 10.5 81 0.0573 611180 35024 593667 6846739 11.2 82 0.0661 463824 30673 448487 4711427 10.2 82 0.0615 576155 35428 558441 6253071 10.9 83 0.0715 433151 30985 417659 4262940 9.8 83 0.0647 540728 34994 523231 5694630 10.5 84 0.0763 402166 30680 386826 3845281 9.6 84 0.0673 505733 34020 488723 5171399 10.2 85 0.0800 371486 29727 356623 3458455 9.3 85 0.0700 471713 33028 455199 4682676 9.9 86 0.0827 341759 28279 327619 3101833 9.1 86 0.0727 438685 31872 422749 4227477 9.6 87 0.0845 313480 26483 300238 2774213 8.8 87 0.0744 406813 30278 391674 3804728 9.4 88 0.0851 286997 24415 274789 2473975 8.6 88 0.0754 376535 28375 362348 3413055 9.1 89 0.0849 262582 22285 251439 2199185 8.4 89 0.0761 348160 26481 334920 3050707 8.8 90 0.0850 240296 20435 230079 1947746 8.1 90 0.0776 321679 24952 309203 2715787 8.4 91 0.0868 219862 19073 210325 1717667 7.8 91 0.0807 296727 23940 284756 2406585 8.1 92 0.0903 200788 18138 191720 1507342 7.5 92 0.0854 272786 23290 261141 2121828 7.8 93 0.0955 182651 17435 173933 1315623 7.2 93 0.0912 249496 22747 238123 1860687 7.5 94 0.1018 165216 16812 156810 1141690 6.9 94 0.0977 226749 22155 215672 1622564 7.2 95 0.1091 148404 16188 140310 984879 6.6 95 0.1049 204595 21462 193864 1406892 6.9 96 0.1174 132216 15518 124457 844569 6.4 96 0.1128 183133 20653 172806 1213029 6.6 97 0.1265 116698 14768 109314 720111 6.2 97 0.1213 162480 19714 152623 1040222 6.4 98 0.1365 101930 13911 94975 610797 6.0 98 0.1305 142766 18630 133451 887600 6.2 99 0.1468 88020 12924 81558 515822 5.9 99 0.1400 124135 17385 115443 754149 6.1 100 0.1571 75096 11797 69197 434265 5.8 100 0.1496 106751 15965 98768 638706 6.0 101 0.1665 63299 10538 58030 365067 5.8 101 0.1584 90785 14379 83596 539938 5.9 102 0.1740 52761 9182 48170 307037 5.8 102 0.1657 76407 12662 70076 456342 6.0 103 0.1787 43579 7788 39686 258867 5.9 103 0.1707 63745 10881 58304 386266 6.1 104 0.1797 35792 6432 32576 219182 6.1 104 0.1726 52864 9127 48300 327962 6.2 105 0.1767 29360 5188 26766 186606 6.4 105 0.1712 43737 7489 39993 279661 6.4 106 0.1702 24171 4114 22114 159840 6.6 106 0.1667 36248 6043 33227 239669 6.6 107 0.1615 20057 3239 18438 137726 6.9 107 0.1600 30205 4833 27789 206442 6.8 108 0.1522 16818 2560 15538 119288 7.1 108 0.1525 25372 3869 23438 178653 7.0 109 0.1440 14258 2053 13231 103750 7.3 109 0.1454 21503 3126 19940 155216 7.2 110 0.1377 12205 1680 11365 90518 7.4 110 0.1395 18377 2564 17095 135276 7.4 111 0.1332 10525 1401 9824 79154 7.5 111 0.1352 15813 2138 14744 118181 7.5 112 0.1301 9123 1187 8530 69330 7.6 112 0.1320 13675 1806 12772 103437 7.6 113 0.1278 7937 1014 7430 60800 7.7 113 0.1296 11870 1539 11100 90664 7.6 114 0.1259 6923 872 6487 53370 7.7 114 0.1276 10331 1319 9671 79564 7.7 115 0.1242 6051 752 5675 46883 7.7 115 0.1258 9012 1133 8445 69893 7.8 116 0.1227 5299 650 4974 41208 7.8 116 0.1239 7879 976 7390 61448 7.8 117 0.1212 4649 563 4367 36234 7.8 117 0.1221 6902 843 6481 54057 7.8 118 0.1198 4086 489 3841 31867 7.8 118 0.1204 6059 729 5695 47577 7.9 119 0.1185 3596 426 3383 28025 7.8 119 0.1187 5330 633 5014 41882 7.9 120 0.1173 3170 372 2984 24642 7.8 120 0.1171 4697 550 4422 36868 7.8 121 0.1162 2799 325 2636 21658 7.7 121 0.1156 4147 479 3908 32446 7.8 122 0.1154 2473 285 2331 19022 7.7 122 0.1143 3668 419 3459 28538 7.8 123 0.1143 2188 250 2063 16691 7.6 123 0.1130 3249 367 3066 25080 7.7 124 0.1135 1938 220 1828 14628 7.5 124 0.1117 2882 322 2721 22014 7.6 125 0.1123 1718 193 1622 12800 7.5 125 0.1109 2560 284 2418 19293 7.5 126 0.1121 1525 171 1440 11179 7.3 126 0.1103 2276 251 2151 16875 7.4 127 0.1115 1354 151 1279 9739 7.2 127 0.1091 2025 221 1915 14725 7.3 128 0.1106 1203 133 1137 8461 7.0 128 0.1086 1804 196 1706 12810 7.1 129 0.1103 1070 118 1011 7324 6.8 129 0.1082 1608 174 1521 11104 6.9 130 0.1103 952 105 900 6313 6.6 130 0.1081 1434 155 1357 9583 6.7 131 0.1098 847 93 801 5414 6.4 131 0.1079 1279 138 1210 8227 6.4 132 0.1088 754 82 713 4613 6.1 132 0.1069 1141 122 1080 7017 6.1 133 0.1086 672 73 636 3900 5.8 133 0.1070 1019 109 965 5937 5.8 134 0.1085 599 65 567 3265 5.4 134 0.1077 910 98 861 4972 5.5 135 0.1086 534 58 505 2698 5.1 135 0.1071 812 87 769 4111 5.1 136 0.1071 476 51 451 2193 4.6 136 0.1076 725 78 686 3343 4.6 137 0.1082 425 46 402 1743 4.1 137 0.1066 647 69 613 2657 4.1 138 0.1082 379 41 359 1341 3.5 138 0.1073 578 62 547 2044 3.5 139 0.1065 338 36 320 982 2.9 139 0.1066 516 55 489 1497 2.9 140 0.1060 302 32 286 662 2.2 140 0.1085 461 50 436 1009 2.2 141 0.1074 270 29 256 376 1.4 141 0.1071 411 44 389 573 1.4 142 1.0000 241 241 121 121 0.5 142 1.0000 367 367 184 183.5 0.5 Appendix B Table B.3 Table B.4 Men Aged 57 in 1992 Women Aged 57 in 1992 x qx lx dx Lx Tx ex x qx lx dx Lx Tx ex

57 0.0198 964461 19142 954890 22373741 23.2 57 0.0173 1016365 17600 1007565 25190927 24.8 58 0.0189 945319 17860 936389 21418851 22.7 58 0.0167 998765 16712 990409 24183363 24.2 59 0.0185 927459 17127 918895 20482462 22.1 59 0.0165 982053 16169 973968 23192954 23.6 60 0.0184 910332 16710 901977 19563567 21.5 60 0.0164 965884 15864 957952 22218986 23.0 61 0.0185 893622 16554 885345 18661590 20.9 61 0.0166 950020 15765 942137 21261034 22.4 62 0.0189 877068 16619 868758 17776245 20.3 62 0.0170 934254 15845 926331 20318897 21.7 63 0.0196 860449 16873 852013 16907487 19.6 63 0.0175 918409 16079 910369 19392565 21.1 64 0.0205 843576 17290 834931 16055475 19.0 64 0.0182 902329 16446 894106 18482196 20.5 65 0.0216 826286 17847 817362 15220544 18.4 65 0.0191 885884 16924 877422 17588090 19.9 66 0.0229 808439 18521 799178 14403181 17.8 66 0.0201 868960 17495 860212 16710668 19.2 67 0.0244 789918 19289 780273 13604003 17.2 67 0.0213 851465 18139 842395 15850456 18.6 68 0.0261 770628 20130 760564 12823730 16.6 68 0.0226 833326 18839 823906 15008061 18.0 69 0.0280 750499 21017 739990 12063166 16.1 69 0.0240 814487 19575 804700 14184155 17.4 70 0.0301 729482 21922 718521 11323176 15.5 70 0.0256 794912 20327 784749 13379455 16.8 71 0.0322 707559 22816 696151 10604655 15.0 71 0.0272 774585 21079 764046 12594707 16.3 72 0.0346 684743 23667 672909 9908504 14.5 72 0.0289 753506 21811 742601 11830661 15.7 73 0.0370 661076 24444 648854 9235595 14.0 73 0.0308 731695 22510 720440 11088060 15.2 74 0.0395 636632 25121 624071 8586741 13.5 74 0.0327 709185 23169 697600 10367620 14.6 75 0.0420 611511 25687 598667 7962670 13.0 75 0.0347 686015 23790 674120 9670020 14.1 76 0.0446 585824 26146 572751 7364002 12.6 76 0.0368 662225 24393 650029 8995900 13.6 77 0.0474 559678 26535 546410 6791251 12.1 77 0.0392 637832 25015 625324 8345871 13.1 78 0.0505 533143 26912 519687 6244841 11.7 78 0.0420 612817 25714 599960 7720547 12.6 79 0.0540 506231 27349 492557 5725154 11.3 79 0.0452 587103 26540 573833 7120587 12.1 80 0.0582 478882 27865 464950 5232597 10.9 80 0.0490 560563 27481 546823 6546754 11.7 81 0.0629 451017 28347 436844 4767647 10.6 81 0.0532 533082 28377 518894 5999932 11.3 82 0.0675 422671 28524 408409 4330803 10.2 82 0.0573 504705 28918 490246 5481038 10.9 83 0.0716 394147 28213 380040 3922394 10.0 83 0.0608 475787 28931 461321 4990792 10.5 84 0.0755 365934 27621 352124 3542354 9.7 84 0.0643 446856 28727 432492 4529471 10.1 85 0.0798 338313 27011 324807 3190230 9.4 85 0.0686 418129 28673 403792 4096978 9.8 86 0.0839 311302 26128 298238 2865423 9.2 86 0.0732 389456 28499 375206 3693186 9.5 87 0.0865 285174 24664 272842 2567185 9.0 87 0.0770 360956 27807 347053 3317980 9.2 88 0.0872 260510 22728 249146 2294344 8.8 88 0.0799 333150 26606 319847 2970927 8.9 89 0.0870 237782 20678 227443 2045198 8.6 89 0.0820 306544 25150 293969 2651080 8.6 90 0.0870 217104 18879 207664 1817755 8.4 90 0.0843 281394 23711 269539 2357111 8.4 91 0.0884 198225 17515 189468 1610091 8.1 91 0.0871 257684 22444 246462 2087572 8.1 92 0.0915 180710 16531 172445 1420623 7.9 92 0.0908 235239 21362 224559 1841110 7.8 93 0.0959 164179 15749 156304 1248179 7.6 93 0.0954 213878 20396 203680 1616551 7.6 94 0.1013 148430 15033 140913 1091874 7.4 94 0.1007 193482 19483 183741 1412871 7.3 95 0.1074 133397 14324 126234 950961 7.1 95 0.1068 173999 18580 164709 1229131 7.1 96 0.1142 119072 13594 112275 824727 6.9 96 0.1136 155420 17652 146593 1064421 6.8 97 0.1215 105479 12816 99071 712451 6.8 97 0.1210 137767 16664 129435 917828 6.7 98 0.1291 92663 11965 86680 613380 6.6 98 0.1287 121103 15586 113310 788393 6.5 99 0.1367 80697 11028 75184 526700 6.5 99 0.1365 105517 14403 98315 675083 6.4 100 0.1436 69670 10005 64667 451517 6.5 100 0.1439 91114 13113 84557 576767 6.3 101 0.1494 59665 8913 55208 386850 6.5 101 0.1504 78001 11734 72134 492210 6.3 102 0.1533 50752 7782 46861 331642 6.5 102 0.1554 66267 10297 61118 420076 6.3 103 0.1549 42970 6656 39642 284781 6.6 103 0.1582 55970 8853 51543 358957 6.4 104 0.1538 36314 5584 33522 245139 6.8 104 0.1583 47117 7460 43387 307414 6.5 105 0.1501 30730 4613 28423 211617 6.9 105 0.1558 39657 6178 36568 264027 6.7 106 0.1446 26117 3776 24229 183193 7.0 106 0.1510 33479 5055 30952 227459 6.8 107 0.1382 22341 3089 20796 158965 7.1 107 0.1448 28425 4116 26367 196507 6.9 108 0.1322 19252 2544 17980 138168 7.2 108 0.1384 24309 3363 22627 170140 7.0 109 0.1271 16708 2123 15646 120188 7.2 109 0.1326 20945 2776 19557 147513 7.0 110 0.1233 14585 1798 13685 104542 7.2 110 0.1279 18169 2324 17007 127956 7.0 111 0.1206 12786 1543 12015 90857 7.1 111 0.1245 15845 1972 14859 110949 7.0 112 0.1188 11244 1336 10576 78842 7.0 112 0.1219 13873 1692 13027 96090 6.9 113 0.1173 9908 1163 9327 68266 6.9 113 0.1200 12181 1462 11450 83063 6.8 114 0.1161 8746 1015 8238 58939 6.7 114 0.1184 10719 1269 10085 71613 6.7 115 0.1148 7731 888 7287 50701 6.6 115 0.1168 9451 1104 8899 61528 6.5 116 0.1136 6843 777 6454 43414 6.3 116 0.1153 8347 962 7865 52630 6.3 117 0.1123 6066 681 5725 36960 6.1 117 0.1138 7384 840 6964 44764 6.1 118 0.1110 5385 598 5086 31234 5.8 118 0.1123 6544 735 6177 37800 5.8 119 0.1098 4787 525 4524 26149 5.5 119 0.1109 5809 644 5487 31623 5.4 120 0.1085 4262 463 4030 21624 5.1 120 0.1095 5165 566 4882 26136 5.1 121 0.1074 3799 408 3595 17594 4.6 121 0.1082 4599 498 4351 21254 4.6 122 0.1063 3391 360 3211 13999 4.1 122 0.1070 4102 439 3882 16904 4.1 123 0.1052 3031 319 2871 10788 3.6 123 0.1059 3663 388 3469 13021 3.6 124 0.1043 2712 283 2570 7917 2.9 124 0.1048 3275 343 3103 9553 2.9 125 0.1034 2429 251 2303 5347 2.2 125 0.1039 2932 305 2779 6449 2.2 126 0.1026 2178 223 2066 3043 1.4 126 0.1030 2627 271 2492 3670 1.4 127 1.0000 1955 1955 977 977 0.5 127 1.0000 2356 2356 1178 1178 0.5 References

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